Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x+2y &= -6 \\ -8x-2y &= -5\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-8x = 2y-5$ Divide both sides by $-8$ to isolate $x$ $x = {-\dfrac{1}{4}y + \dfrac{5}{8}}$ Substitute this expression for $x$ in the first equation. $5({-\dfrac{1}{4}y + \dfrac{5}{8}}) + 2y = -6$ $-\dfrac{5}{4}y + \dfrac{25}{8} + 2y = -6$ Simplify by combining terms, then solve for $y$ $\dfrac{3}{4}y + \dfrac{25}{8} = -6$ $\dfrac{3}{4}y = -\dfrac{73}{8}$ $y = -\dfrac{73}{6}$ Substitute $-\dfrac{73}{6}$ for $y$ in the top equation. $5x+2( -\dfrac{73}{6}) = -6$ $5x-\dfrac{73}{3} = -6$ $5x = \dfrac{55}{3}$ $x = \dfrac{11}{3}$ The solution is $\enspace x = \dfrac{11}{3}, \enspace y = -\dfrac{73}{6}$.